Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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516 views

Fourier transform of the Cantor function

Let $f:[0,1] \to [0,1]$ be the Cantor function. Extend $f$ to all of $\mathbb R$ by setting $f(x)=0$ on $\mathbb R \setminus [0,1]$. Calculate the Fourier transform of $f$ $$ \hat f(x)= \int f(t) ...
16
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1answer
483 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: ...
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1answer
45 views

The Fourier transform of $x \mapsto |x|^{-1/3}e^{-x^{2}}$ is not in $L^{1}$.

Okay previously my lecturer showed that this is so by proving in the following way: Proof by contradiction. Suppose the transform is in $L^{1}$. Then as $f \in L^1$, we may use Fourier Inversion ...
3
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1answer
91 views

$f \in L^1(\mathbb R), f>0$ then $|\hat f(y)| < \hat f(0), y \ne 0$

Suppose $f$ is a strictly positive function in $L^1(\mathbb R)$. Show $$ |\hat f(y)| < \hat f(0) \text{, for all } y \ne 0. $$ Using monotonicity of the integral, I can show $|\hat f(y)| \le \hat ...
4
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1answer
150 views

Multiplicative formula for Fourier transforms

Suppose I have two functions continuous functions $f,g \in \mathcal{S}(\Bbb{R})$, the Schwartz space. Now I know that the following multiplicative formula holds. Namely, if $\hat{g}$ denotes the ...
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1answer
413 views

What is the Fourier transformation of a uniform B-Spline?

I'm looking for the Fourier transformation of the (constant) uniform B-Spline $$N_0(x) = \begin{cases}1 & 0 \leqslant x < 1 \\ 0 & otherwise \end{cases}$$ If $N_0(x)$ would also attain ...
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1answer
91 views

How to show that for arbitrary( complex) trigonometric polynomials $P$ and $Q$ holds…

How to show that for arbitrary( complex) trigonometric polynomials $P$ and $Q$ holds $$\frac{1}{2\pi} \int_{-\pi}^{\pi} P(t)Q(mt)dt = \frac{1}{2\pi} \int_{-\pi}^{\pi} P(t)dt \frac{1}{2\pi} ...
3
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1answer
90 views

If the Fourier series of $f$ is absolutely convergent does it implies that it converges to f

If $$ \sum_{k=-\infty}^{\infty} |\hat f (k) | < \infty $$ does it implies $$ S_n(t)=\sum_{k=-n}^{n} \hat f (k) e^{ikt} \to f(t) \; ? $$ I know $S_n$ converges for each $t$ to some function $S$. ...
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0answers
50 views

ways to prove that $e^{i|\xi|}\phi$ is a fourier transform of $L^1$ function

Let $\phi\in C_{0}^{\infty}(\mathbb{R}^n)$ and equals to 1 near the origin,then show that $e^{i|\xi|}\phi(\frac{|\xi|}{\mu})$ is a fourier transform of an $L^1$ function with any $\mu>0$,and how ...
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139 views

Sobolev-type inequality.

Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
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1answer
23 views

How to change variables if you only know the invariant measure

I want to find an equation relating the coefficients of two different harmonic expansions of the same function (and a relation between their respective basis functions). ...
0
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1answer
103 views

How to formulate arbitrary complex trigonometric polynomial?

How to formulate arbitrary complex trigonometric polynomial? I know that in real form it is $\displaystyle\sum_{n=1}^k a_n\cos(nx)+b_n\sin(nx)$
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1answer
506 views

Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence ...
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1answer
81 views

Fourier transform and sampling time

Given a signal $x(t)$ and the $X(\omega)$ obtained from $x(t)$ using a FFT with a sampling time $Ts$, I get a subset of $X(\omega)$: $Y(\omega)$ obtained from $X(\omega)$ taking it between $\omega_0$ ...
2
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0answers
331 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
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1answer
51 views

$f$ absolutely continuous, $\hat f(n) \downarrow 0$, $\hat f(n)$ positive and even, then $\sum \hat f_n <\infty$

Suppose $f$ is absolutely continuous and the Fourier coefficients of $f$ satisfy $$ \hat f(n)= \frac{a_n sgn(n)}{n} \ge 0 $$ where the $a_n$ are postive, even, and decreasing to zero as $|n| \to ...
3
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1answer
250 views

If $f$ is absolutely continuous then $\sum|\hat f (n) | < \infty$

Let $f$ be a $2\pi$ periodic function on $[0,2\pi]$. If $f$ is absolutely continuous, is it true that the sum of its Fourier coefficient converges absolutely $$ S(t)=\sum_{n=-\infty}^{\infty} | \hat f ...
0
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1answer
142 views

Fourier Series with Signals

So the question is: Determine the fourier series representations for the following signal: Here the formula for the fourier series $$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} ...
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3answers
454 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
0
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1answer
67 views

$ f \in W^{s,2}$ then $ \int_{\Bbb R^n} \xi_j^{2s} | \mathscr F f( \xi) |^2 d \xi < \infty $?

If $f \in W^{s,2} (\Bbb R^n) $, then by the Plancherel's theorem, I know that its Fourier transform $ \mathscr F f(\xi) \in L^2 (\Bbb R^n) $. ($ \scr F$ means the Fourier transform). Now I want to ...
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1answer
164 views

Calculating the Fourier transform of $f(t)=Ae^{-i\omega_0 t}$

I am trying to calculate the Fourier transform of $f(t)=Ae^{-i\omega_0 t}$ I'm getting an infinity which is giving me problems. Here are my steps: ...
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1answer
1k views

Fourier transform of a piecewise function

I am trying to find the Fourier transform of $$f(x)=Ae^{-\alpha|x|}$$ where $\alpha>0$. $f(x)$ becomes an even piecewise function defined over the intervals $-\infty$ to $0$ and $0$ to $\infty$. ...
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1answer
54 views

Show from every positive sequence decreasing to zero, we can build a sequence satisfying a concavity condition

Let $\{x_n\}_{n=-\infty}^{\infty}$ be a positive sequence decreasing to zero as $|n| \to \infty$. Show there is a sequence $\{y_n\}$ satisfying \begin{align} y_n >& x_n \tag{1}\\ ...
2
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1answer
100 views

About the continuity of the Fourier transform.

If $ x^{\alpha} g \in L^1 ( \Bbb R^n)$ for $| \alpha | \leqslant k$, then how can I prove that its Fourier transform $$ \mathscr{F} g \in C^k ( \Bbb R^n) ?$$ Here $\alpha$ is a multi-index.
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96 views

Show $\left\lvert \frac{1}{\pi+2} \sum_{0 \le k \le 2n,k\ne n} \frac{i}{k-n} e^{ikt} \right\rvert > c \ln n, c>0$

The partial sum of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ can be written as \begin{align} S_n(f,t) &= \sum_{0<|k| \le n} \frac{i}{k} e^{ikt} \; (1) \\ & = ...
2
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1answer
141 views

Fourier transform of characteristic function in a sphere

A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...
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1answer
122 views

How to show that $\frac{1}{\tan(x/2)}=2 \sum_{j=1}^{\infty}\sin(jx)$ in Cesàro way/sense?

Show that if $x \neq 0,\pm 2 \pi,\pm 4 \pi, \dots$, then $$\frac{1}{\tan(x/2)}=2 \sum_{j=1}^{\infty}\sin(jx)$$ in Cesàro way/sense. Some hint whether to manipulate ...
2
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1answer
144 views

Convergence of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$

The partial sum of the Fourier series for the function $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ is $$ S_n(t)=-2 \sum_{k=1}^{n} \frac{\sin kt}{k} $$ We saw a theorem which states that the Fourier ...
0
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1answer
85 views

Equivalence of two ways of expressing Fourier transform

One way of expressing Fourier transform would be $(\mathcal{F}f)(t)=\int_{-\infty}^\infty f(x)\, e^{-itx}\,dx$ and its inversion $f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty (\mathcal{F}f)(t)\, ...
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0answers
134 views

Discrete fourier transform and Frobenius norm

I have thousands of data files which are essential DFT data of plots such as this: And a DFT of the plot with a "Hard" threshold of 0.9 gives me: This DFT is just the left top corner of the DFT ...
0
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1answer
221 views

Show $\left\lvert\sum_{k=-n}^n \frac{\sin k t }{k}\right\rvert \le \pi + 2$ for all $n$ and $t$, $t \in [0,2\pi]$

Let $f(t)=(t-\pi)\chi_{(0,2\pi)}$, $t \in [0,2\pi]$, then the partial sum of the Fourier series of $f$ is $$ S_n(t)=- \sum_{0 < |k| \le n} \frac{\sin k t}{k}. $$ Show $|S_n(t)| \le \pi+2$ for all ...
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1answer
237 views

Help me please, A0 fourier series for this Sawtooth wave

I need to know for this wave sub zero, the correct answer is a0=-2V/Pi, but I do not know how to get to that answer. Help me please the imagen its the series. This is the sawtooth wave ...
4
votes
2answers
185 views

Howto show that function is a representation fot the delta function via complex path integrals?

So given is the definition: $$ f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{ikx}dk $$ I'm supposed to show that this is a representation of the Dirac delta "function" ($f(x) = \delta(x)$) ...
2
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0answers
104 views

Calculating a Poisson probability from the chacteristic function?

In a previous homework assignment we were given a function that corresponds to an arbitrary angular distribution $A_{FB}=(F-B)/(F+B)=(F-B)/N$, where F = # of events in the forward hemisphere, B = # of ...
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1answer
170 views

The maximum absolute value of DFT of window vector

Let x=[1, ⋯ ,1, 0, ⋯ ,0] be a window vector of length N, which consists of B consecutive 1s and the remaining N-B consecutive 0s. I took the N-point DFT on x and got X=[X_0, X_1, ⋯, X_(N-1)] which is ...
3
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0answers
256 views

How is study of fractals related to fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
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1answer
50 views

how to prove the convolution formular?

let $\overset{\backsim} {g}(x)=g(-x)$; suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove ...
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1answer
30 views

variables of Fourier analysis - how to prove their relations

Fourier transform: $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\ e^{- 2\pi i t \xi}\,dx$ where $t$ can be time and $\xi$ can be frequency. So, the question is how do we prove that $t$ and $\xi$ can in ...
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0answers
34 views

Proof that every real function has negative frequency component

I want to know why every real function has negative frequency component. If I am not correct, can anyone tell me how it is really? I heard that it is related to Fourier analysis, though not sure. ...
2
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0answers
155 views

Discete Fourier Transform-Additive Combinatorics

Although I am not completely familiar with the subject but I have met two 'dual' definitions of the Discrete Fourier Transform of a function $ f: \mathbb{Z} / N \mathbb{Z} \rightarrow \mathbb{C} $ , ...
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1answer
173 views

Show $\exists f \in C \ni \|f\|_\infty \le 1$, but it's Fourier series diverges

Show $\exists f \in C \ni \|f\|_\infty \le 1$, but it's Fourier series diverges. The proof is in our textbook (Katznelson, Harmonic analysis). It uses this argument. Let ...
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0answers
124 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
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0answers
92 views

Filter/Removal of periodic 'delta-peaks'

Currently I am measuring data (Counts over Time). Due to measurement problems I have some nasty peaks in this data. These peaks are periodical, very sharp (~3 datapoints over a range of 10000) and ...
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3answers
501 views

convergence of autocorrelation function and existence of Fourier transform

I am studying the Wiener-Khinchin Theorem. But, I am wondering why the Dirichlet condition, which says that the autocorrelation function of WSS should be absolutely integrable, is sufficient for the ...
3
votes
1answer
177 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
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1answer
89 views

Fourier transform, why this gives incorrect answer?

Let $f(x) = \begin{cases}e^{-x} & ,0<x<1\\0 & ,\text{Otherwise}\end{cases}$ I'm trying to calculate the fourier transform of $xf(x)$, by using the fact that $xf(x) = -\frac{d}{da}f(a ...
0
votes
1answer
309 views

Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem. If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$. How to prove it?
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2answers
181 views

Finding spectrum using the convolution property

Using the convolution property, find the spectrum for $$w(t)= \sin(2\pi f_1 t) \cos(2\pi f_2 t).$$ I'm confused on how to solve this question. Can you give me any aproach?
2
votes
1answer
355 views

A question on a solution of an inhomogeneous heat equation.

I am now working on the following PDE equation (Evan's PDE textbook Section 2.5 No.14) \begin{align} u_{t}-\Delta u + cu=f \ \ & on \ \ \mathbb{R}^n\times (0,\infty) \\ u=g \ \ & on \ \ ...
6
votes
2answers
330 views

Paley-Wiener type theorems for distributions?

In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of ...